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Really want to know? Math follows…(high school algebra level, but still).Einstein worked from two postulates, both of which had been confirmed many times by experiment, and have since been repeatedly confirmed by experiment.There is no preferred “frame of reference” in the Universe; all motion is relative. This was Galileo’s Principle of Relativity, which he expressed beautifully as Galileo's shipShut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still.Note that you are doing the Galileo’s Ship experiment even as you read this! You are moving at a dizzying variety of velocities right now. See: How Fast Are You Moving When You Are Standing Still?, for example. In physics, this translates into “velocity is meaningless except with respect to a frame of reference” — I can’t say “the ball is moving at 60 meters per second”; I have to say “the ball is moving at 60 meters per second with respect to me”. And this is our common experience: if you are moving away from me at 30 meters per second, and I throw a ball after you at 60 meters per second, you’ll see the ball move towards you at 30 meters per second. A consequence of this is that velocity cannot appear in physical law. We cannot, for example, say that light moves at speed c; we have to say that light moves at speed c with respect to the laboratory apparatus.Light moves at exactly c.You read that right. Not “[math]c[/math] with respect to the laboratory apparatus”. [math]c[/math]. If you are moving away from me at 30 meters per second, and I send a light beam after you, you will see the beam coming towards you at exactly [math]c[/math], not [math]c - 30[/math].If you think those two postulates are in direct contradiction, you are exactly right. But both had been confirmed many times by experiment. Postulate #2 had come from Maxwell's equations, which (still today) are the foundation of electromagnetism, and had been confirmed by the Michelson–Morley experiment and by the Fizeau experiment.So both postulates were true, and they were in direct contradiction.Einstein realized that the only way these could both be true was if moving observers measured time and space differently, so he calculated the map. Einstein sat down to consider the following question. Suppose Alice is moving at speed [math]v[/math] with respect to Bob. Alice measures the space-time coordinates of events as [math](x, t)[/math]. Bob measures them as [math](X, T)[/math]. For convenience, we assume they agree on the point [math](0, 0)[/math]. Given Bob’s measurement [math](X, T)[/math], of an event, can I determine Alice’s measurement [math](x, t)[/math]?Einstein knew the transformation had to be linear, so let’s start with the linear equations:[math]x = aX + bT[/math][math]t = eX + fT[/math]We must also have, of course[math]X = ax + bt[/math][math]T = ex + ft[/math]where [math]a, b, e, f[/math] are some functions of [math]v[/math] to be determined later. Since Alice is moving at speed [math]v[/math] with respect to Bob, Alice’s coordinates in Bob’s spacetime are on the line [math](vT, T)[/math]; in Alice’s they are on the axis [math]x = 0[/math], so[math]vT = bt[/math][math]T = ft[/math]Applying high-school algebra, [math]vft = bt, b = vf[/math]. Notice immediately that [math]v = b/f[/math], so [math]f \neq 0[/math], or [math]v[/math] would be infinite.Postulate two says that if Alice measures an event at [math](c, 1)[/math], Bob measures it at [math](cT, T)[/math]. Putting these in:[math]cT = ac + b[/math][math]T = ec + f[/math]Thus[math]ac + b = c(ec + f)[/math]Also, if Alice measures an event at [math](-c, 1)[/math], Bob measures it at [math](-cT, T)[/math], so[math]cT = ac - b[/math][math]T = f - ec[/math][math]c(f - ec) = ac - b[/math]We now have two equations:[math]ac + b = ec^2 + fc[/math][math]ac - b = fc - ec^2[/math]If we add them together, we get:[math]2ac = 2c f[/math][math]a = f[/math]If we subtract them, we get[math]2b = 2ec^2, b = ec^2, vf = ec^2, e = vf/c^2[/math]We now have everything in terms of a single variable, [math]f[/math][math]x = cfX + vfT[/math][math]t = \frac{vf}{c^2} X + fT[/math]And all we need to do is determine [math]f[/math]. As it happens, the conventional symbol for what I’ve been calling [math]f[/math] is [math]\gamma[/math], so I’ll rewrite this[math]x = \gamma(X + vT)[/math]Now, the inverse transformation is[math]X = \gamma(x - vt)[/math]substituting[math]x = \gamma(\gamma(x - vt) + vT)[/math]or[math]\gamma v T = x - \gamma^2 x + \gamma^2 v t[/math]or[math]T = \gamma t + x\frac{1 - \gamma^2}{v \gamma}[/math]If we have a beam of light, this will have [math] X = cT[/math], so[math]\gamma(x - vt) = c (\gamma t + x \frac{1 - \gamma^2}{v \gamma})[/math][math]\gamma x - \gamma v t = c \gamma t + cx \frac{1 - \gamma^2}{v \gamma}[/math][math]v \gamma^2 x - \gamma^2 v^2 t = v c \gamma^2 t + c x (1 - \gamma^2)[/math][math]x (v \gamma^2 + c \gamma^2 - c) = t \gamma^2 v (c + v)[/math]For the light cone, [math]x = ct[/math][math]ct (v \gamma^2 + c \gamma^2 - c) = t \gamma^2 v (c + v)[/math][math]c(v \gamma^2 + c \gamma^2 - c) = \gamma^2 v (c + v)[/math][math]cv \gamma^2 + c^2 \gamma^2 - c^2 = \gamma^2 vc + v^2 \gamma^2[/math][math]\gamma^2 (c^2 - v^2) = c^2[/math][math]\gamma^2 = \frac{c^2}{c^2 - v^2}[/math][math]\gamma^2 = \frac{1}{1 - \frac{v^2}{c^2}}[/math][math]\gamma = \sqrt{\frac{1}{1 - \frac{v^2}{c^2}}}[/math]Ta da! We have the Lorentz Transformation . It had previously been suggested by Henrik Lorentz, on the basis of geometric considerations, to explain the Michelson-Morley experiment, but without any real foundation. Einstein’s contribution was to show that it was a necessary consequence of the two postulates, which (I cannot say this enough) had been found to be true by experiment.This derivation, by the way, is only a slightly simplified form of Einstein’s.